Globally positive solutions for the third order differential equation with the damping term and delay, $$x^{\text{'}\text{'}\text{'}}+q\left(t\right)x^{\text{'}}\left(t\right)-r\left(t\right)f\left(x\left(\phi \left(t\right)\right)\right)=0,$$ are studied in the case where the corresponding second order differential equation $$y^{\text{'}\text{'}}+q\left(t\right)y=0$$ is oscillatory. Necessary and sufficient conditions for all nonoscillatory solutions of (*) to be unbounded are given. Furthermore, oscillation criteria ensuring that any solution is either oscillatory or unbounded together with its first and second derivatives are presented. The comparison of results with those...

In this paper, we aim to study the global solvability of the following system of third order nonlinear neutral delay differential equations $$\frac{d}{dt}\left\{r_i\left(t\right)\frac{d}{dt}\left[{\lambda }_i\left(t\right)\frac{d}{dt}\left(x_i\left(t\right)-f_i(t,x_1(t-{\sigma }_{i1}),x_2(t-{\sigma }_{i2}),x_3(t-{\sigma }_{i3}))\right)\right]\right\}+\frac{d}{dt}\left[r_i\left(t\right)\frac{d}{dt}{g}_i(t,x_1\left(p_{i1}\left(t\right)\right),x_2\left(p_{i2}\left(t\right)\right),x_3\left(p_{i3}\left(t\right)\right))\right]+\frac{d}{dt}{h}_i(t,x_1\left(q_{i1}\left(t\right)\right),x_2\left(q_{i2}\left(t\right)\right),x_3\left(q_{i3}\left(t\right)\right))=l_i(t,x_1\left({\eta }_{i1}\left(t\right)\right),x_2\left({\eta }_{i2}\left(t\right)\right),x_3\left({\eta }_{i3}\left(t\right)\right)),t\ge t_0,i\in \{1,2,3\}$$ in the following bounded closed and convex set $$\Omega (a,b)=\left\{x\left(t\right)=(x_1\left(t\right),x_2\left(t\right),x_3\left(t\right))\in C([t_0,+\infty ),{ℝ}^3):a\left(t\right)\le x_i\left(t\right)\le b\left(t\right),\forall t\ge t_0,i\in \{1,2,3\}\right\},$$ where ${\sigma }_{ij}>0$, $r_i,{\lambda }_i,a,b\in C([t_0,+\infty ),{ℝ}^{+})$, $f_i,g_i,h_i,l_i\in C([t_0,+\infty )\times {ℝ}^3,ℝ)$, $p_{ij},q_{ij},{\eta }_{ij}\in C([t_0,+\infty ),ℝ)$ for $i,j\in \{1,2,3\}$. By applying the Krasnoselskii fixed point theorem, the Schauder fixed point theorem, the Sadovskii fixed point theorem and the Banach contraction principle, four existence results of uncountably many bounded positive solutions of the system are established.